To find a line that is perpendicular to the given line \( y = -74x \) and passes through the point \((-7, 5)\), we need to follow these steps:
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Determine the slope of the given line: The slope of the line \( y = -74x \) is \(-74\).
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Find the slope of the perpendicular line: The slope of a line that is perpendicular to another is the negative reciprocal of the original line's slope. Thus, the slope \( m \) of the perpendicular line is: \[ m = -\frac{1}{-74} = \frac{1}{74} \]
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Use the point-slope form to find the equation of the perpendicular line: The point-slope form of a line is given by: \[ y - y_1 = m(x - x_1) \] Substituting in the point \((-7, 5)\) and the slope \(\frac{1}{74}\): \[ y - 5 = \frac{1}{74}(x + 7) \]
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Simplify the equation: \[ y - 5 = \frac{1}{74}x + \frac{7}{74} \] \[ y = \frac{1}{74}x + \frac{7}{74} + 5 \]
To combine \( \frac{7}{74} + 5 \), convert 5 into seventies: \[ 5 = \frac{370}{74} \] Therefore: \[ y = \frac{1}{74}x + \frac{7}{74} + \frac{370}{74} = \frac{1}{74}x + \frac{377}{74} \]
Now we need to identify which of the options matches our result \( y = \frac{1}{74}x + \frac{377}{74} \):
None of the given equations correspond exactly to the result of our calculations. However, if there were an option among the responses that mentioned having a slope of \( \frac{1}{74} \) or was structured closely to that form, we would choose it.
Upon closer scrutiny based on the options provided, it appears that there's a mistake. The closest equivalent format among those provided might either be incorrect or requires reconsideration of arithmetic transformations due to matching formats.
However, if we're strictly preventing from mixing signs or transformations based on the given answers, take special note of answers and correctness to maintain \( y = mx + b \) form. Ideally, since they wanted perpendicular and we've reasoned through the slope correctly, you can conclude the real match on paper.
In conclusion, strive to ensure through confirmation between formats and expand potential misconceptions on slopes which could be misrepresented through the options. Hence \( y = \frac{1}{74}x + b\) matters more regarding closeness in terms.
If remaining open to human error in interpretation, strictly follow the perpendicularity and retain basis over calculations via continuous checks through specified conditions like slopes matching more apparent terms.
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